Mathematicians From The United States

"The world of antitrust is reminiscent of Alice’s Wonderland: everything seemingly is, yet apparently isn’t, simultaneously. It is a world in which competition is lauded as the basic axiom and guiding principle, yet "too much" competition is condemned as "cutthroat." It is a world in which actions designed to limit competition are branded as criminal when taken by businessmen, yet praised as "enlightened" when initiated by the government. It is a world in which the law is so vague that businessmen have no way of knowing whether specific actions will be declared illegal until they hear the judge’s verdict—after the fact."

"Once stock prices reach the point at which it is hard to value them by logical methodology, stocks will be bought as they were in the late 1920s not for investment but to be unloaded at a still higher price. The ensuing break could be disastrous because panic psychology cannot be summarily altered or reversed by easing money policies."

"Without calling the overall national issue a bubble, it's pretty clear that it's an unsustainable underlying pattern."

"While local economies may experience significant price imbalances, a national severe price distortion seems most unlikely in the United States, given its size and diversity."

"Rising interest rates have been advertised for so long and in so many places that anyone who has not appropriately hedged this position by now obviously is desirous of losing money."

"American consumers might benefit if lenders provided greater mortgage product alternatives to the traditional fixed-rate mortgage."

"Intensive research in recent years into the sources of economic growth among both developing and developed nations generally point to a number of important factors: the state of knowledge and skill of a population; the degree of control over indigenous natural resources; the quality of a country's legal system, particularly a strong commitment to a rule of law and protection of property rights; and yes, the extent of a country's openness to trade with the rest of the world. For the United States, arguably the most important factor is the type of rule of law under which economic activity takes place. When asked abroad why the United States has become the most prosperous large economy in the world, I respond, with only mild exaggeration, that our forefathers wrote a constitution and set in motion a system of laws that protects individual rights, especially the right to own property. Nonetheless, the degree of state protection is sometimes in dispute. But by and large, secure property rights are almost universally accepted by Americans as a critical pillar of our economy. While the right of property in the abstract is generally uncontested in all societies embracing democratic market capitalism, different degrees of property protection do apparently foster different economic incentives and outcomes."

"It hardly makes any difference who will be the next president. The world is governed by market forces."

"The reason there is very little support for the gold standard is the consequences of those types of market adjustments are not considered to be appropriate in the 20th and 21st century. I am one of the rare people who have still some nostalgic view about the old gold standard, as you know, but I must tell you, I am in a very small minority among my colleagues on that issue."

"Nor can private counterparties restrict supplies of gold, another commodity whose derivatives are often traded over-the-counter, where central banks stand ready to lease gold in increasing quantities should the price rise."

"For some, the benign inflation outcome of 1996 might be considered surprising, as resource utilization rates--particularly of labor--were in the neighborhood of those that historically have been associated with building inflation pressures. To be sure, an acceleration in nominal labor compensation, especially its wage component, became evident over the past year. But the rate of pay increase still was markedly less than historical relationships with labor market conditions would have predicted. Atypical restraint on compensation increases has been evident for a few years now and appears to be mainly the consequence of greater worker insecurity. In 1991, at the bottom of the recession, a survey of workers at large firms by International Survey Research Corporation indicated that 25 percent feared being laid off. In 1996, despite the sharply lower unemployment rate and the tighter labor market, the same survey organization found that 46 percent were fearful of a job layoff."

"Linear programming is viewed as a revolutionary development giving man the ability to state general objectives and to find, by means of the simplex method, optimal policy decisions for a broad class of practical decision problems of great complexity. In the real world, planning tends to be ad hoc because of the many special-interest groups with their multiple objectives."

"In retrospect... it is interesting to note that the original problem that started my research is still outstanding -- namely the problem of planning or scheduling dynamically over time, particularly planning dynamically under uncertainty. If such a problem could be successfully solved it could eventually through better planning contribute to the well-being and stability of the world."

"During my first year at Berkeley I arrived late one day to one of Neyman's classes. On the blackboard were two problems which I assumed had been assigned for homework. I copied them down. A few days later I apologized to Neyman for taking so long to do the homework - the problems seemed to be a little harder to do than usual. I asked him if he still wanted the work. He told me to throw it on his desk. I did so reluctantly because his desk was covered with such a heap of papers that I feared my homework would be lost there forever. About six weeks later, one Sunday morning about eight o'clock, Anne and I were awakened by someone banging on our front door. It was Neyman. He rushed in with papers in hand, all excited: "I've just written an introduction to one of your papers. Read it so I can send it out right away for publication." For a minute I had no idea what he was talking about. To make a long story short, the problems on the blackboard which I had solved thinking they were homework were in fact two famous unsolved problems in statistics. That was the first inkling I had that there was anything special about them."

"The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight."

"One of the first applications of the simplex algorithm was to the determination of an adequate diet that was of least cost. In the fall of 1947, Jack Laderman of the Mathematical Tables Project of the National Bureau of Standards undertook, as a test of the newly proposed simplex method, the first large-scale computation in this field. It was a system with nine equations in seventy-seven unknowns. Using hand-operated desk calculators, approximately 120 man-days were required to obtain a solution. … The particular problem solved was one which had been studied earlier by George Stigler (who later became a Nobel Laureate) who proposed a solution based on the substitution of certain foods by others which gave more nutrition per dollar. He then examined a "handful" of the possible 510 ways to combine the selected foods. He did not claim the solution to be the cheapest but gave his reasons for believing that the cost per annum could not be reduced by more than a few dollars. Indeed, it turned out that Stigler's solution (expressed in 1945 dollars) was only 24 cents higher than the true minimum per year $39.69."

"If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming."

"Industrial production, the flow of resources in the economy, the exertion of military effort in a war theater-all are complexes of numerous interrelated activities. Differences may exist in the goals to be achieved, the particular processes involved, and the magnitude of effort. Nevertheless, it is possible to abstract the underlying essential similarities in the management of these seemingly disparate systems."

"All such problems can be formulated as mathematical programming problems. Naturally, we can propose many sophisticated algorithms and a theory but the final test of a theory is its capacity to solve the problems which originated it."

"Industrial production, the flow of resources in the economy, the exertion of military effort in a war, the management of finances --all require the coordination of interrelated activities. What these complex undertakings share in common is the task of constructing a statement of actions to be performed, their timing and quantity (called a program or schedule), that, if implemented, would move the system from a given initial status as much as possible towards some defined goal"

"In all the works on pedagogy that ever I read — and they have been many, big, and heavy — I don't remember that any one has advocated a system of teaching by practical jokes, mostly cruel. That, however, describes the method of our great teacher, Experience. She says,"

"Professor Klein then speaks of "that artistic finish that we admire in Euclid's Elements," and mentions Allman's important historical work. I heartily concur in this estimate of Euclid, and desire to contrast it with the error of Charles S. Peirce, in the Nation, where he speaks of "Euclid's proof (Elements Bk. I., props. 16 and 17)" as "really quite fallacious, because it uses no premises not as true in the case of spherics." Our bright American seems to have forgotten Euclid's Postulate 6 (Axiom 12 in Gregory, Axiom 9 in Heiberg), "Two straight lines cannot enclose a space;" that is, two straights having crossed never recur."

"Conceptual graphs are system of logic based on the existential graphs of Charles Sanders Peirce and the semantic networks of artificial intelligence. The purpose of the system is to express meaning in a form that is logically precise, humanly readable, and computationally tractable."

"Ultimately, the last word is never ours. Peirce said: the meaning of my life is entrusted to others. [...] Which proves that truth is never something definitive; we are always mistaken. On a journey. It is no coincidence that I speak of the “transit of truth.”"

"One of the greatest philosophers of all time."

"Beyond doubt ... he was one of the most original minds of the later nineteenth century and certainly the greatest American thinker ever."

"Peirce was a man of tremendous energy, producing a multitude of ideas, good, bad, and indifferent. He reminds on of a volcano spouting vast masses of rock, of which some, on examination, turn out to be nuggets of pure gold. He holds―and I confess that an examination of scientific inference has made me feel the force of this view―that man is adapted, by his congenital constitution, to the apprehension of natural laws which cannot be proved by experience, although experience is in conformity with them."

"Peirce held that the continuity of space, time, ideation, feeling, and perception is an irreducible deliverance of science, and that an adequate conception of the continuous is an extremely important part of all the sciences. This doctrine he called “synechism,” a word deriving from the Greek preposition that means “(together) with.”"

"James called Peirce the most original thinker of their generation; Peirce placed himself somewhere near the rank of Leibniz. This much is now certain; he is the most original and versatile of America's philosophers and America's greatest logician."

"Peirce was concerned to explicate the idea of meaning whereas James was concerned to explicate the meaning of ideas."

"The principle of interpretation says that "a sign is something by knowing which we know something more" (Peirce). The Peircean idea of semiosis is the idea of an infinite process of interpretation."

"Pierce wrote as a logician and James as a humanist."

"Peirce, to my knowledge, is original and unique in stressing the problem of studying the rules that limit the class of possible theories. Of course, his concept of abduction, like Lorenz’s biological a priori, has a strongly Kantian flavor, and all derive from the rationalist psychology that concerned itself with the forms, the limits, and the principles that provide “the sinews and connections” for human thought, that underlie “that infinite amount of knowledge of which we are not always conscious,” of which Leibnitz spoke. It is therefore quite natural that we should link these developments to the revival of philosophical grammar, which grew from the same soil as an attempt, quite fruitful and legitimate, to explore one basic facet of human intelligence."

"Our whole past experience is continually in our consciousness, though most of it sunk to a great depth of dimness. I think of consciousness as a bottomless lake, whose waters seem transparent, yet into which we can clearly see but a little way."

"Mere imagination would indeed be mere trifling; only no imagination is mere."

"All the evolution we know of proceeds from the vague to the definite."

"Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object."

"Let us not pretend to doubt in philosophy what we do not doubt in our hearts."

"To say, therefore, that thought cannot happen in an instant, but requires a time, is but another way of saying that every thought must be interpreted in another, or that all thought is in signs."

"Every man is fully satisfied that there is such a thing as truth, or he would not ask any question."

"It is ...easy to be certain. One has only to be sufficiently vague."

"The ordinary logic has a great deal to say about genera and species, or in our nineteeth century dialect, about classes. Now a class is a set of objects comprising all that stand to one another in a special relation of similarity. But where ordinary logic talks of classes the logic of relatives talks of systems. A system is a set of objects comprising all that stands to one another in a group of connected relations. Induction according to ordinary logic rises from the contemplation of a sample of a class to that of a whole class; but according to the logic of relatives it rises from the comtemplation of a fragment of a system to the envisagement of the complete system."

"Effort supposes resistance."

"The idea does not belong to the soul; it is the soul that belongs to the idea."

"Do not block the way of inquiry."

"If we endeavor to form our conceptions upon history and life, we remark three classes of men. The first consists of those for whom the chief thing is the qualities of feelings. These men create art. The second consists of the practical men, who carry on the business of the world. They respect nothing but power, and respect power only so far as it [is] exercized. The third class consists of men to whom nothing seems great but reason. If force interests them, it is not in its exertion, but in that it has a reason and a law. For men of the first class, nature is a picture; for men of the second class, it is an opportunity; for men of the third class, it is a cosmos, so admirable, that to penetrate to its ways seems to them the only thing that makes life worth living."

"My original essay, having been written for a popular monthly, assumes, for no better reason than that real inquiry cannot begin until a state of real doubt arises and ends as soon as Belief is attained, that "a settlement of Belief," or, in other words, a state of satisfaction, is all that Truth, or the aim of inquiry, consists in. The reason I gave for this was so flimsy, while the inference was so nearly the gist of Pragmaticism, that I must confess the argument of that essay might with some justice be said to beg the question. The first part of the essay, however, is occupied with showing that, if Truth consists in satisfaction, it cannot be any actual satisfaction, but must be the satisfaction which would ultimately be found if the inquiry were pushed to its ultimate and indefeasible issue. This, I beg to point out, is a very different position from that of Mr Schiller and the pragmatists of to-day. I trust I shall be believed when I say that it is only a desire to avoid being misunderstood in consequence of my relations with pragmatism, and by no means as arrogating any superior immunity from error which I have too good reason to know that I do not enjoy, that leads me to express my personal sentiments about their tenets. Their avowedly undefinable position, if it be not capable of logical characterisation, seems to me to be characterised by an angry hatred of strict logic, and even some disposition to rate any exact thought which interferes with their doctrines as all humbug. At the same time, it seems to me clear that their approximate acceptance of the Pragmaticist principle, and even that very casting aside of difficult distinctions (although I cannot approve of it), has helped them to a mightily clear discernment of some fundamental truths that other philosophers have seen but through a mist, and most of them not at all. Among such truths — all of them old, of course, yet acknowledged by few — I reckon their denial of necessitarianism; their rejection of any "consciousness" different from a visceral or other external sensation; their acknowledgment that there are, in a Pragmatistical sense, Real habits (which Really would produce effects, under circumstances that may not happen to get actualised, and are thus Real generals); and their insistence upon interpreting all hypostatic abstractions in terms of what they would or might (not actually will) come to in the concrete. It seems to me a pity they should allow a philosophy so instinct with life to become infected with seeds of death in such notions as that of the unreality of all ideas of infinity and that of the mutability of truth, and in such confusions of thought as that of active willing (willing to control thought, to doubt, and to weigh reasons) with willing not to exert the will (willing to believe)."

"The third argument, enclosing and defending the other two, consists in the development of those principles of logic according to which the humble argument is the first stage of a scientific inquiry into the origin of the three Universes, but of an inquiry which produces, not merely scientific belief, which is always provisional, but also a living, practical belief, logically justified in crossing the Rubicon with all the freightage of eternity."

"The difference between a pessimistic and an optimistic mind is of such controlling importance in regard to every intellectual function, and especially for the conduct of life, that it is out of the question to admit that both are normal, and the great majority of mankind are naturally optimistic."

"Unless man have a natural bent in accordance with nature's, he has no chance of understanding nature at all."